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Maccoll's Modes of Modalities Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences 15-1 | 2011 Hugh MacColl after One Hundred Years MacColl’s Modes of Modalities Fabien Schang Publisher Editions Kimé Electronic version Printed version URL: http:// Date of publication: 1 avril 2011 philosophiascientiae.revues.org/510 Number of pages: 149-188 DOI: 10.4000/philosophiascientiae.510 ISBN: 978-2-84174-551-7 ISSN: 1775-4283 ISSN: 1281-2463 Electronic reference Fabien Schang, « MacColl’s Modes of Modalities », Philosophia Scientiæ [Online], 15-1 | 2011, Online since 01 April 2014, connection on 30 September 2016. URL : http:// philosophiascientiae.revues.org/510 ; DOI : 10.4000/philosophiascientiae.510 The text is a facsimile of the print edition. Tous droits réservés MacColl’s Modes of Modalities Fabien Schang Technische Universität Dresden, Institut für Philosophie, Germany LHSP, Archives H. Poincaré (UMR 7117), Nancy-Université, France Résumé : Hugh MacColl est présenté d’ordinaire comme un pionnier des logiques modales et multivalentes, suite à son introduction de modalités qui vont au-delà de la simple vérité et fausseté. Mais un examen plus attentif montre que cet héritage est discutable et devrait tenir compte de la façon dont ces modalités procédaient. Bien que MacColl ait conçu une logique modale au sens large du terme, nous montrerons qu’il n’a pas produit une logique multivalente au sens strict. Sa logique serait comparable plutôt à une « logique non-fregéenne », c’est-à-dire une logique algébrique qui effectue une partition au sein de la classe des vérités et faussetés mais n’étend pas pour autant le domaine des valeurs de vérité. Abstract: Hugh MacColl is commonly seen as a pioneer of modal and many- valued logic, given his introduction of modalities that go beyond plain truth and falsehood. But a closer examination shows that such a legacy is debatable and should take into account the way in which these modalities proceeded. We argue that, while MacColl devised a modal logic in the broad sense of the word, he did not give rise to a many-valued logic in the strict sense. Rather, his logic is similar to a “non-Fregean logic”: an algebraic logic that partitions the semantic classes of truth and falsehood into subclasses but does not extend the range of truth-values. Modalities and many-valuedness A preliminary attention to the notation is in order. MacColl’s Logic (hereafter: MCL) resorts to a symbolic language that is in accordance with the algebraic style of George Boole or Ernst Schröder. Mathematical signs are used to characterize operations between any propositions AB and CD. Thus, AB + CD stands for their sum, ABCD (or AB×CD, or AB.CD) for their product, AB ∶ CD for the implication from AB to CD, AB = CD for their equivalence Philosophia Scientiæ, 15 (1), 2011, 163–188. 164 Fabien Schang B D D B B B (synonymous with (A ∶ C )(C ∶ A )), and (A )′ for the denial of A . MCL is thus a logic of predication that consists in propositions subsuming a class under another one: in AB, the first term A is the subject term while the second term B is the predicate term whose application results in a proposition such as “A is B”, or “The A is a B”. A first terminological distinction is that between the occurrence of propo- sitions as terms or factors and as exponents. AB and CD occur as terms in the sum AB +CD and as factors in the product ABCD. Also, A and C are the subjects of the predications AB and CD; while B and D occur as exponents, because they are predicates to which A and C respectively belong. Another distinction is that between the adjectival and predicative use: B is used ad- B C jectivally in AB , and predicatively in A , so that the whole proposition AB means that the A which is a B is also a C. On the one hand, this symbolic device helps to convey information about quantity by introducing integers into the adjectival and predicative uses. Thus A1, A2 ..., An specify n different individuals in the class A, and the superscript 0 0 gives an expression to quantification in terms of the null class: AB means that -0 no A is B (“the A that is B belongs to the null class”), and AB that some A is 0 B (“the A that is B does not belong to the null class”); A-B means that every -0 A is B (“the A that is not B belongs to the null class”), and A-B that some A is not B (“the A that is not B does not belong to the null class”). On the other hand, the later distinction between metalanguage and object-language doesn’t appear in MCL: the truth-values are not confined to a separate higher- order language and are treated on a par with any other term of the symbolic logic. The result is a class of semantic predicates that go beyond truth and falsehood, as stated by MacColl: The symbol Aτ only asserts that A is true in a particular case or instance. The symbol Aε asserts more than this: it asserts that A is certain, that A is always true (or true in every case) within the limits of our data and definitions, that its probability is 1. The symbol Aι only asserts that A is false in a particular case or instance; it says nothing as to the truth or falsehood of A in other instances. The symbol Aη asserts more than this; it asserts that A contradicts some datum or definition, that its probability is 0. Thus Aτ and Aι are simply assertive; each refers to one case, and raises no general question as to data or probability. The symbol Aθ (A is a variable) is equivalent to A-ηA-ε; it asserts that A is neither impossible nor certain, that is, that A is possible but uncertain. In other words, Aθ asserts that the probability of A is neither 0 nor 1, but some proper fraction between the two. [MacColl 1906, 7] MacColl supplements the five preceding semantic predicates with four other elements, i.e. π for possibility, p for probability, q for improbability, MacColl’s modes of modalities 165 and u for uncertainty [MacColl 1906, 14]. Most of these predicates (except for p and q) behave like modalities, since they are reminiscent of the Aristotelian logic and its isomorphic oppositions within quantified and modal propositions: certainty (“every A is B”) and impossibility (“every A is not B”) correspond to universal affirmation and negation, while possibility (“some A is B”) and uncertainty (“some A is not B”) correspond to particular affirmation and nega- tion. As to the remaining cases of truth, falsehood, variability, probability and improbability, they require an extension of the Aristotelian square: variability (“some A is B , some A is not B”) corresponds to two-sided possibility or con- tingency, and such a concept finds its rightful place in the first logical hexagon of Robert Blanché [Blanché 1953]; truth (“this A is B”) and falsehood (“this A is not B”) correspond not to universal or particular, but singular propositions, whose rightful place requires further extension from the second logical hexagon of Tadeusz Czeżowski [Czeżowski 1955] to the logical octagon of Jan Woleński [Woleński 1998] (see figures [1–4]). 166 Fabien Schang The result is the following list of 28 logical oppositions and their five kinds of opposition: 1 Contraries (5): {ε, η}; {ε,θ}; {η,θ}; {η, τ}; {ε,ι} Subcontraries (5): {π,u}; {π, -θ}; {u, -θ}; {π,ι}; {u, τ} Contradictories (4): {ε,u}; {η,π}; {-θ,θ}; {τ,ι} Subalterns (10): {ε,π}; {η,u}; {θ,π}; {θ,u}; {ε, -θ}; {η, -θ}; {ε, τ}; {η,ι}; {τ,π}; {ι,u} Mere non-contradictories (4): {-θ,ι}; {-θ, τ}; {ι,θ}; {τ,θ} The 28 oppositions above exhaust the set of logical relations between MacColl’s modalities, excluding the peculiar cases of p and q: the latter are probable values that cannot be located within a polygon of oppositions, given that they represent any of the indefinitely many values between full truth (or certainty: 1) and full falsehood (or impossibility: 0). The dotted lines stand for contradictory relations; the arrows indicate the entailment relation between an antecedent A and its consequent B, such that A ∶ B is a theorem. Here is a sample of those theorems from MacColl [MacColl 1906, 8]: (T12) (Aτ + Aι) expresses the law of excluded middle between contradictory terms, (T14) (Aε + Aη + Aθ )ε means that three modalities exhaust the semantic class of MCL, while the implications (T15) Aε ∶ Aτ and (T16) Aη ∶ Aι correspond to subalternation relations. 1 and 0 are usually used to symbolize truth and falsehood, in algebraic se- mantics; but it has been frequently claimed by the commentators on MacColl (including [Simons 1998]) that he thought of truth-values in probabilistic terms of truth-cases. If so, then any of the logical values of MCL should be reformu- lated within the closed interval [0,1] of an infinitely-valued logic. This should not be a proper characterization of MacColl’s view of modalities, however: the difference between particular and singular judgments cannot be rendered within such a probabilistic line, and the above oppositions depict logical rela- tions that are qualitative rather than quantitative (as witnessed by the absence of p and q from the polygons). At any rate, it may seem strange to state logical oppositions between concepts expressing truth-values: if an opposition is about the possibility for two terms to be both true or false, what does it mean for τ and ι not to be true or false together? A plausible answer is that these alleged truth-values are not that but, rather, modal operators attached to proposi- tions.
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